MATL, 16 bytes

tZv=Gq:"t5BZ+]vs 

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###Explanation

This repeatedly applies convolution to generate the rows. For example, for input n=5 we start with the first row

0 0 0 0 1 0 0 0 0 

Convolving with [1 0 1] gives

0 0 0 1 0 1 0 0 0 

Repeating the operation gives

0 0 1 0 2 0 1 0 0 

then

0 1 0 3 0 3 0 1 0 

etc. Concatenating these arrays vertically and computing the sum of each column gives the result.

t % Input n implictly. Duplicate Zv % Symmetric range. Gives [1 2 3 4 5 4 3 2 1] for input 5 = % Equal to (element-wise). Gives [0 0 0 0 1 0 0 0 0]. This is the first row Gq: % Push [1 2 ... n-1] " % For each. This executes the following code n-1 times t % Duplicate 5B % Push 5 in binary, that is, [1 0 1] Z+ % Convolution keeping size ] % End v % Concatenate all results vertically s % Sum. Display implicitly. 

MATL, 16 bytes

tZv=Gq:"t5BZ+]vs 

Try it online!

Explanation

This repeatedly applies convolution to generate the rows. For example, for input n=5 we start with the first row

0 0 0 0 1 0 0 0 0 

Convolving with [1 0 1] gives

0 0 0 1 0 1 0 0 0 

Repeating the operation gives

0 0 1 0 2 0 1 0 0 

then

0 1 0 3 0 3 0 1 0 

etc. Concatenating these arrays vertically and computing the sum of each column gives the result.

t % Input n implictly. Duplicate Zv % Symmetric range. Gives [1 2 3 4 5 4 3 2 1] for input 5 = % Equal to (element-wise). Gives [0 0 0 0 1 0 0 0 0]. This is the first row Gq: % Push [1 2 ... n-1] " % For each. This executes the following code n-1 times t % Duplicate 5B % Push 5 in binary, that is, [1 0 1] Z+ % Convolution keeping size ] % End v % Concatenate all results vertically s % Sum. Display implicitly. 

MATL, 16 bytes

tZv=Gq:"t5BZ+]vs 

Try it online!

###Explanation

This repeatedly applies convolution to generate the rows. For example, for input n=5 we start with the first row

0 0 0 0 1 0 0 0 0 

Convolving with [1 0 1] gives

0 0 0 1 0 1 0 0 0 

Repeating the operation gives

0 0 1 0 2 0 1 0 0 

then

0 1 0 3 0 3 0 1 0 

etc.

t % Input n implictly. Duplicate Zv % Symmetric range. Gives [1 2 3 4 5 4 3 2 1] for input 5 = % Equal to (element-wise). Gives [0 0 0 0 1 0 0 0 0]. This is the first row Gq: % Push [1 2 ... n-1] " % For each. This executes the following code n-1 times t % Duplicate 5B % Push 5 in binary, that is, [1 0 1] Z+ % Convolution keeping size ] % End v % Concatenate all results vertically s % Sum. Display implicitly. 

MATL, 16 bytes

tZv=Gq:"t5BZ+]vs 

Try it online!

###Explanation

This repeatedly applies convolution to generate the rows. For example, for input n=5 we start with the first row

0 0 0 0 1 0 0 0 0 

Convolving with [1 0 1] gives

0 0 0 1 0 1 0 0 0 

Repeating the operation gives

0 0 1 0 2 0 1 0 0 

then

0 1 0 3 0 3 0 1 0 

etc. Concatenating these arrays vertically and computing the sum of each column gives the result.

t % Input n implictly. Duplicate Zv % Symmetric range. Gives [1 2 3 4 5 4 3 2 1] for input 5 = % Equal to (element-wise). Gives [0 0 0 0 1 0 0 0 0]. This is the first row Gq: % Push [1 2 ... n-1] " % For each. This executes the following code n-1 times t % Duplicate 5B % Push 5 in binary, that is, [1 0 1] Z+ % Convolution keeping size ] % End v % Concatenate all results vertically s % Sum. Display implicitly. 

MATL, 16 bytes

tZv=Gq:"t5BZ+]vs 

Try it online!

MATL, 16 bytes

tZv=Gq:"t5BZ+]vs 

Try it online!

###Explanation

This repeatedly applies convolution to generate the rows. For example, for input n=5 we start with the first row

0 0 0 0 1 0 0 0 0 

Convolving with [1 0 1] gives

0 0 0 1 0 1 0 0 0 

Repeating the operation gives

0 0 1 0 2 0 1 0 0 

then

0 1 0 3 0 3 0 1 0 

etc.

t % Input n implictly. Duplicate Zv % Symmetric range. Gives [1 2 3 4 5 4 3 2 1] for input 5 = % Equal to (element-wise). Gives [0 0 0 0 1 0 0 0 0]. This is the first row Gq: % Push [1 2 ... n-1] " % For each. This executes the following code n-1 times t % Duplicate 5B % Push 5 in binary, that is, [1 0 1] Z+ % Convolution keeping size ] % End v % Concatenate all results vertically s % Sum. Display implicitly.